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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.util;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import java.math.BigInteger;<a name="line.19"></a>
<FONT color="green">020</FONT>    import java.util.concurrent.atomic.AtomicReference;<a name="line.20"></a>
<FONT color="green">021</FONT>    <a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.MathArithmeticException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.NotPositiveException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.exception.util.Localizable;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.26"></a>
<FONT color="green">027</FONT>    <a name="line.27"></a>
<FONT color="green">028</FONT>    /**<a name="line.28"></a>
<FONT color="green">029</FONT>     * Some useful, arithmetics related, additions to the built-in functions in<a name="line.29"></a>
<FONT color="green">030</FONT>     * {@link Math}.<a name="line.30"></a>
<FONT color="green">031</FONT>     *<a name="line.31"></a>
<FONT color="green">032</FONT>     * @version $Id: ArithmeticUtils.java 1422313 2012-12-15 18:53:41Z psteitz $<a name="line.32"></a>
<FONT color="green">033</FONT>     */<a name="line.33"></a>
<FONT color="green">034</FONT>    public final class ArithmeticUtils {<a name="line.34"></a>
<FONT color="green">035</FONT>    <a name="line.35"></a>
<FONT color="green">036</FONT>        /** All long-representable factorials */<a name="line.36"></a>
<FONT color="green">037</FONT>        static final long[] FACTORIALS = new long[] {<a name="line.37"></a>
<FONT color="green">038</FONT>                           1l,                  1l,                   2l,<a name="line.38"></a>
<FONT color="green">039</FONT>                           6l,                 24l,                 120l,<a name="line.39"></a>
<FONT color="green">040</FONT>                         720l,               5040l,               40320l,<a name="line.40"></a>
<FONT color="green">041</FONT>                      362880l,            3628800l,            39916800l,<a name="line.41"></a>
<FONT color="green">042</FONT>                   479001600l,         6227020800l,         87178291200l,<a name="line.42"></a>
<FONT color="green">043</FONT>               1307674368000l,     20922789888000l,     355687428096000l,<a name="line.43"></a>
<FONT color="green">044</FONT>            6402373705728000l, 121645100408832000l, 2432902008176640000l };<a name="line.44"></a>
<FONT color="green">045</FONT>    <a name="line.45"></a>
<FONT color="green">046</FONT>        /** Stirling numbers of the second kind. */<a name="line.46"></a>
<FONT color="green">047</FONT>        static final AtomicReference&lt;long[][]&gt; STIRLING_S2 = new AtomicReference&lt;long[][]&gt; (null);<a name="line.47"></a>
<FONT color="green">048</FONT>    <a name="line.48"></a>
<FONT color="green">049</FONT>        /** Private constructor. */<a name="line.49"></a>
<FONT color="green">050</FONT>        private ArithmeticUtils() {<a name="line.50"></a>
<FONT color="green">051</FONT>            super();<a name="line.51"></a>
<FONT color="green">052</FONT>        }<a name="line.52"></a>
<FONT color="green">053</FONT>    <a name="line.53"></a>
<FONT color="green">054</FONT>        /**<a name="line.54"></a>
<FONT color="green">055</FONT>         * Add two integers, checking for overflow.<a name="line.55"></a>
<FONT color="green">056</FONT>         *<a name="line.56"></a>
<FONT color="green">057</FONT>         * @param x an addend<a name="line.57"></a>
<FONT color="green">058</FONT>         * @param y an addend<a name="line.58"></a>
<FONT color="green">059</FONT>         * @return the sum {@code x+y}<a name="line.59"></a>
<FONT color="green">060</FONT>         * @throws MathArithmeticException if the result can not be represented<a name="line.60"></a>
<FONT color="green">061</FONT>         * as an {@code int}.<a name="line.61"></a>
<FONT color="green">062</FONT>         * @since 1.1<a name="line.62"></a>
<FONT color="green">063</FONT>         */<a name="line.63"></a>
<FONT color="green">064</FONT>        public static int addAndCheck(int x, int y)<a name="line.64"></a>
<FONT color="green">065</FONT>                throws MathArithmeticException {<a name="line.65"></a>
<FONT color="green">066</FONT>            long s = (long)x + (long)y;<a name="line.66"></a>
<FONT color="green">067</FONT>            if (s &lt; Integer.MIN_VALUE || s &gt; Integer.MAX_VALUE) {<a name="line.67"></a>
<FONT color="green">068</FONT>                throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, x, y);<a name="line.68"></a>
<FONT color="green">069</FONT>            }<a name="line.69"></a>
<FONT color="green">070</FONT>            return (int)s;<a name="line.70"></a>
<FONT color="green">071</FONT>        }<a name="line.71"></a>
<FONT color="green">072</FONT>    <a name="line.72"></a>
<FONT color="green">073</FONT>        /**<a name="line.73"></a>
<FONT color="green">074</FONT>         * Add two long integers, checking for overflow.<a name="line.74"></a>
<FONT color="green">075</FONT>         *<a name="line.75"></a>
<FONT color="green">076</FONT>         * @param a an addend<a name="line.76"></a>
<FONT color="green">077</FONT>         * @param b an addend<a name="line.77"></a>
<FONT color="green">078</FONT>         * @return the sum {@code a+b}<a name="line.78"></a>
<FONT color="green">079</FONT>         * @throws MathArithmeticException if the result can not be represented as an<a name="line.79"></a>
<FONT color="green">080</FONT>         *         long<a name="line.80"></a>
<FONT color="green">081</FONT>         * @since 1.2<a name="line.81"></a>
<FONT color="green">082</FONT>         */<a name="line.82"></a>
<FONT color="green">083</FONT>        public static long addAndCheck(long a, long b) throws MathArithmeticException {<a name="line.83"></a>
<FONT color="green">084</FONT>            return ArithmeticUtils.addAndCheck(a, b, LocalizedFormats.OVERFLOW_IN_ADDITION);<a name="line.84"></a>
<FONT color="green">085</FONT>        }<a name="line.85"></a>
<FONT color="green">086</FONT>    <a name="line.86"></a>
<FONT color="green">087</FONT>        /**<a name="line.87"></a>
<FONT color="green">088</FONT>         * Returns an exact representation of the &lt;a<a name="line.88"></a>
<FONT color="green">089</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.89"></a>
<FONT color="green">090</FONT>         * Coefficient&lt;/a&gt;, "{@code n choose k}", the number of<a name="line.90"></a>
<FONT color="green">091</FONT>         * {@code k}-element subsets that can be selected from an<a name="line.91"></a>
<FONT color="green">092</FONT>         * {@code n}-element set.<a name="line.92"></a>
<FONT color="green">093</FONT>         * &lt;p&gt;<a name="line.93"></a>
<FONT color="green">094</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.94"></a>
<FONT color="green">095</FONT>         * &lt;ul&gt;<a name="line.95"></a>
<FONT color="green">096</FONT>         * &lt;li&gt; {@code 0 &lt;= k &lt;= n } (otherwise<a name="line.96"></a>
<FONT color="green">097</FONT>         * {@code IllegalArgumentException} is thrown)&lt;/li&gt;<a name="line.97"></a>
<FONT color="green">098</FONT>         * &lt;li&gt; The result is small enough to fit into a {@code long}. The<a name="line.98"></a>
<FONT color="green">099</FONT>         * largest value of {@code n} for which all coefficients are<a name="line.99"></a>
<FONT color="green">100</FONT>         * {@code  &lt; Long.MAX_VALUE} is 66. If the computed value exceeds<a name="line.100"></a>
<FONT color="green">101</FONT>         * {@code Long.MAX_VALUE} an {@code ArithMeticException} is<a name="line.101"></a>
<FONT color="green">102</FONT>         * thrown.&lt;/li&gt;<a name="line.102"></a>
<FONT color="green">103</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.103"></a>
<FONT color="green">104</FONT>         *<a name="line.104"></a>
<FONT color="green">105</FONT>         * @param n the size of the set<a name="line.105"></a>
<FONT color="green">106</FONT>         * @param k the size of the subsets to be counted<a name="line.106"></a>
<FONT color="green">107</FONT>         * @return {@code n choose k}<a name="line.107"></a>
<FONT color="green">108</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.108"></a>
<FONT color="green">109</FONT>         * @throws NumberIsTooLargeException if {@code k &gt; n}.<a name="line.109"></a>
<FONT color="green">110</FONT>         * @throws MathArithmeticException if the result is too large to be<a name="line.110"></a>
<FONT color="green">111</FONT>         * represented by a long integer.<a name="line.111"></a>
<FONT color="green">112</FONT>         */<a name="line.112"></a>
<FONT color="green">113</FONT>        public static long binomialCoefficient(final int n, final int k)<a name="line.113"></a>
<FONT color="green">114</FONT>            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {<a name="line.114"></a>
<FONT color="green">115</FONT>            ArithmeticUtils.checkBinomial(n, k);<a name="line.115"></a>
<FONT color="green">116</FONT>            if ((n == k) || (k == 0)) {<a name="line.116"></a>
<FONT color="green">117</FONT>                return 1;<a name="line.117"></a>
<FONT color="green">118</FONT>            }<a name="line.118"></a>
<FONT color="green">119</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.119"></a>
<FONT color="green">120</FONT>                return n;<a name="line.120"></a>
<FONT color="green">121</FONT>            }<a name="line.121"></a>
<FONT color="green">122</FONT>            // Use symmetry for large k<a name="line.122"></a>
<FONT color="green">123</FONT>            if (k &gt; n / 2) {<a name="line.123"></a>
<FONT color="green">124</FONT>                return binomialCoefficient(n, n - k);<a name="line.124"></a>
<FONT color="green">125</FONT>            }<a name="line.125"></a>
<FONT color="green">126</FONT>    <a name="line.126"></a>
<FONT color="green">127</FONT>            // We use the formula<a name="line.127"></a>
<FONT color="green">128</FONT>            // (n choose k) = n! / (n-k)! / k!<a name="line.128"></a>
<FONT color="green">129</FONT>            // (n choose k) == ((n-k+1)*...*n) / (1*...*k)<a name="line.129"></a>
<FONT color="green">130</FONT>            // which could be written<a name="line.130"></a>
<FONT color="green">131</FONT>            // (n choose k) == (n-1 choose k-1) * n / k<a name="line.131"></a>
<FONT color="green">132</FONT>            long result = 1;<a name="line.132"></a>
<FONT color="green">133</FONT>            if (n &lt;= 61) {<a name="line.133"></a>
<FONT color="green">134</FONT>                // For n &lt;= 61, the naive implementation cannot overflow.<a name="line.134"></a>
<FONT color="green">135</FONT>                int i = n - k + 1;<a name="line.135"></a>
<FONT color="green">136</FONT>                for (int j = 1; j &lt;= k; j++) {<a name="line.136"></a>
<FONT color="green">137</FONT>                    result = result * i / j;<a name="line.137"></a>
<FONT color="green">138</FONT>                    i++;<a name="line.138"></a>
<FONT color="green">139</FONT>                }<a name="line.139"></a>
<FONT color="green">140</FONT>            } else if (n &lt;= 66) {<a name="line.140"></a>
<FONT color="green">141</FONT>                // For n &gt; 61 but n &lt;= 66, the result cannot overflow,<a name="line.141"></a>
<FONT color="green">142</FONT>                // but we must take care not to overflow intermediate values.<a name="line.142"></a>
<FONT color="green">143</FONT>                int i = n - k + 1;<a name="line.143"></a>
<FONT color="green">144</FONT>                for (int j = 1; j &lt;= k; j++) {<a name="line.144"></a>
<FONT color="green">145</FONT>                    // We know that (result * i) is divisible by j,<a name="line.145"></a>
<FONT color="green">146</FONT>                    // but (result * i) may overflow, so we split j:<a name="line.146"></a>
<FONT color="green">147</FONT>                    // Filter out the gcd, d, so j/d and i/d are integer.<a name="line.147"></a>
<FONT color="green">148</FONT>                    // result is divisible by (j/d) because (j/d)<a name="line.148"></a>
<FONT color="green">149</FONT>                    // is relative prime to (i/d) and is a divisor of<a name="line.149"></a>
<FONT color="green">150</FONT>                    // result * (i/d).<a name="line.150"></a>
<FONT color="green">151</FONT>                    final long d = gcd(i, j);<a name="line.151"></a>
<FONT color="green">152</FONT>                    result = (result / (j / d)) * (i / d);<a name="line.152"></a>
<FONT color="green">153</FONT>                    i++;<a name="line.153"></a>
<FONT color="green">154</FONT>                }<a name="line.154"></a>
<FONT color="green">155</FONT>            } else {<a name="line.155"></a>
<FONT color="green">156</FONT>                // For n &gt; 66, a result overflow might occur, so we check<a name="line.156"></a>
<FONT color="green">157</FONT>                // the multiplication, taking care to not overflow<a name="line.157"></a>
<FONT color="green">158</FONT>                // unnecessary.<a name="line.158"></a>
<FONT color="green">159</FONT>                int i = n - k + 1;<a name="line.159"></a>
<FONT color="green">160</FONT>                for (int j = 1; j &lt;= k; j++) {<a name="line.160"></a>
<FONT color="green">161</FONT>                    final long d = gcd(i, j);<a name="line.161"></a>
<FONT color="green">162</FONT>                    result = mulAndCheck(result / (j / d), i / d);<a name="line.162"></a>
<FONT color="green">163</FONT>                    i++;<a name="line.163"></a>
<FONT color="green">164</FONT>                }<a name="line.164"></a>
<FONT color="green">165</FONT>            }<a name="line.165"></a>
<FONT color="green">166</FONT>            return result;<a name="line.166"></a>
<FONT color="green">167</FONT>        }<a name="line.167"></a>
<FONT color="green">168</FONT>    <a name="line.168"></a>
<FONT color="green">169</FONT>        /**<a name="line.169"></a>
<FONT color="green">170</FONT>         * Returns a {@code double} representation of the &lt;a<a name="line.170"></a>
<FONT color="green">171</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.171"></a>
<FONT color="green">172</FONT>         * Coefficient&lt;/a&gt;, "{@code n choose k}", the number of<a name="line.172"></a>
<FONT color="green">173</FONT>         * {@code k}-element subsets that can be selected from an<a name="line.173"></a>
<FONT color="green">174</FONT>         * {@code n}-element set.<a name="line.174"></a>
<FONT color="green">175</FONT>         * &lt;p&gt;<a name="line.175"></a>
<FONT color="green">176</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.176"></a>
<FONT color="green">177</FONT>         * &lt;ul&gt;<a name="line.177"></a>
<FONT color="green">178</FONT>         * &lt;li&gt; {@code 0 &lt;= k &lt;= n } (otherwise<a name="line.178"></a>
<FONT color="green">179</FONT>         * {@code IllegalArgumentException} is thrown)&lt;/li&gt;<a name="line.179"></a>
<FONT color="green">180</FONT>         * &lt;li&gt; The result is small enough to fit into a {@code double}. The<a name="line.180"></a>
<FONT color="green">181</FONT>         * largest value of {@code n} for which all coefficients are &lt;<a name="line.181"></a>
<FONT color="green">182</FONT>         * Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,<a name="line.182"></a>
<FONT color="green">183</FONT>         * Double.POSITIVE_INFINITY is returned&lt;/li&gt;<a name="line.183"></a>
<FONT color="green">184</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.184"></a>
<FONT color="green">185</FONT>         *<a name="line.185"></a>
<FONT color="green">186</FONT>         * @param n the size of the set<a name="line.186"></a>
<FONT color="green">187</FONT>         * @param k the size of the subsets to be counted<a name="line.187"></a>
<FONT color="green">188</FONT>         * @return {@code n choose k}<a name="line.188"></a>
<FONT color="green">189</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.189"></a>
<FONT color="green">190</FONT>         * @throws NumberIsTooLargeException if {@code k &gt; n}.<a name="line.190"></a>
<FONT color="green">191</FONT>         * @throws MathArithmeticException if the result is too large to be<a name="line.191"></a>
<FONT color="green">192</FONT>         * represented by a long integer.<a name="line.192"></a>
<FONT color="green">193</FONT>         */<a name="line.193"></a>
<FONT color="green">194</FONT>        public static double binomialCoefficientDouble(final int n, final int k)<a name="line.194"></a>
<FONT color="green">195</FONT>            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {<a name="line.195"></a>
<FONT color="green">196</FONT>            ArithmeticUtils.checkBinomial(n, k);<a name="line.196"></a>
<FONT color="green">197</FONT>            if ((n == k) || (k == 0)) {<a name="line.197"></a>
<FONT color="green">198</FONT>                return 1d;<a name="line.198"></a>
<FONT color="green">199</FONT>            }<a name="line.199"></a>
<FONT color="green">200</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.200"></a>
<FONT color="green">201</FONT>                return n;<a name="line.201"></a>
<FONT color="green">202</FONT>            }<a name="line.202"></a>
<FONT color="green">203</FONT>            if (k &gt; n/2) {<a name="line.203"></a>
<FONT color="green">204</FONT>                return binomialCoefficientDouble(n, n - k);<a name="line.204"></a>
<FONT color="green">205</FONT>            }<a name="line.205"></a>
<FONT color="green">206</FONT>            if (n &lt; 67) {<a name="line.206"></a>
<FONT color="green">207</FONT>                return binomialCoefficient(n,k);<a name="line.207"></a>
<FONT color="green">208</FONT>            }<a name="line.208"></a>
<FONT color="green">209</FONT>    <a name="line.209"></a>
<FONT color="green">210</FONT>            double result = 1d;<a name="line.210"></a>
<FONT color="green">211</FONT>            for (int i = 1; i &lt;= k; i++) {<a name="line.211"></a>
<FONT color="green">212</FONT>                 result *= (double)(n - k + i) / (double)i;<a name="line.212"></a>
<FONT color="green">213</FONT>            }<a name="line.213"></a>
<FONT color="green">214</FONT>    <a name="line.214"></a>
<FONT color="green">215</FONT>            return FastMath.floor(result + 0.5);<a name="line.215"></a>
<FONT color="green">216</FONT>        }<a name="line.216"></a>
<FONT color="green">217</FONT>    <a name="line.217"></a>
<FONT color="green">218</FONT>        /**<a name="line.218"></a>
<FONT color="green">219</FONT>         * Returns the natural {@code log} of the &lt;a<a name="line.219"></a>
<FONT color="green">220</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.220"></a>
<FONT color="green">221</FONT>         * Coefficient&lt;/a&gt;, "{@code n choose k}", the number of<a name="line.221"></a>
<FONT color="green">222</FONT>         * {@code k}-element subsets that can be selected from an<a name="line.222"></a>
<FONT color="green">223</FONT>         * {@code n}-element set.<a name="line.223"></a>
<FONT color="green">224</FONT>         * &lt;p&gt;<a name="line.224"></a>
<FONT color="green">225</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.225"></a>
<FONT color="green">226</FONT>         * &lt;ul&gt;<a name="line.226"></a>
<FONT color="green">227</FONT>         * &lt;li&gt; {@code 0 &lt;= k &lt;= n } (otherwise<a name="line.227"></a>
<FONT color="green">228</FONT>         * {@code IllegalArgumentException} is thrown)&lt;/li&gt;<a name="line.228"></a>
<FONT color="green">229</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.229"></a>
<FONT color="green">230</FONT>         *<a name="line.230"></a>
<FONT color="green">231</FONT>         * @param n the size of the set<a name="line.231"></a>
<FONT color="green">232</FONT>         * @param k the size of the subsets to be counted<a name="line.232"></a>
<FONT color="green">233</FONT>         * @return {@code n choose k}<a name="line.233"></a>
<FONT color="green">234</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.234"></a>
<FONT color="green">235</FONT>         * @throws NumberIsTooLargeException if {@code k &gt; n}.<a name="line.235"></a>
<FONT color="green">236</FONT>         * @throws MathArithmeticException if the result is too large to be<a name="line.236"></a>
<FONT color="green">237</FONT>         * represented by a long integer.<a name="line.237"></a>
<FONT color="green">238</FONT>         */<a name="line.238"></a>
<FONT color="green">239</FONT>        public static double binomialCoefficientLog(final int n, final int k)<a name="line.239"></a>
<FONT color="green">240</FONT>            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {<a name="line.240"></a>
<FONT color="green">241</FONT>            ArithmeticUtils.checkBinomial(n, k);<a name="line.241"></a>
<FONT color="green">242</FONT>            if ((n == k) || (k == 0)) {<a name="line.242"></a>
<FONT color="green">243</FONT>                return 0;<a name="line.243"></a>
<FONT color="green">244</FONT>            }<a name="line.244"></a>
<FONT color="green">245</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.245"></a>
<FONT color="green">246</FONT>                return FastMath.log(n);<a name="line.246"></a>
<FONT color="green">247</FONT>            }<a name="line.247"></a>
<FONT color="green">248</FONT>    <a name="line.248"></a>
<FONT color="green">249</FONT>            /*<a name="line.249"></a>
<FONT color="green">250</FONT>             * For values small enough to do exact integer computation,<a name="line.250"></a>
<FONT color="green">251</FONT>             * return the log of the exact value<a name="line.251"></a>
<FONT color="green">252</FONT>             */<a name="line.252"></a>
<FONT color="green">253</FONT>            if (n &lt; 67) {<a name="line.253"></a>
<FONT color="green">254</FONT>                return FastMath.log(binomialCoefficient(n,k));<a name="line.254"></a>
<FONT color="green">255</FONT>            }<a name="line.255"></a>
<FONT color="green">256</FONT>    <a name="line.256"></a>
<FONT color="green">257</FONT>            /*<a name="line.257"></a>
<FONT color="green">258</FONT>             * Return the log of binomialCoefficientDouble for values that will not<a name="line.258"></a>
<FONT color="green">259</FONT>             * overflow binomialCoefficientDouble<a name="line.259"></a>
<FONT color="green">260</FONT>             */<a name="line.260"></a>
<FONT color="green">261</FONT>            if (n &lt; 1030) {<a name="line.261"></a>
<FONT color="green">262</FONT>                return FastMath.log(binomialCoefficientDouble(n, k));<a name="line.262"></a>
<FONT color="green">263</FONT>            }<a name="line.263"></a>
<FONT color="green">264</FONT>    <a name="line.264"></a>
<FONT color="green">265</FONT>            if (k &gt; n / 2) {<a name="line.265"></a>
<FONT color="green">266</FONT>                return binomialCoefficientLog(n, n - k);<a name="line.266"></a>
<FONT color="green">267</FONT>            }<a name="line.267"></a>
<FONT color="green">268</FONT>    <a name="line.268"></a>
<FONT color="green">269</FONT>            /*<a name="line.269"></a>
<FONT color="green">270</FONT>             * Sum logs for values that could overflow<a name="line.270"></a>
<FONT color="green">271</FONT>             */<a name="line.271"></a>
<FONT color="green">272</FONT>            double logSum = 0;<a name="line.272"></a>
<FONT color="green">273</FONT>    <a name="line.273"></a>
<FONT color="green">274</FONT>            // n!/(n-k)!<a name="line.274"></a>
<FONT color="green">275</FONT>            for (int i = n - k + 1; i &lt;= n; i++) {<a name="line.275"></a>
<FONT color="green">276</FONT>                logSum += FastMath.log(i);<a name="line.276"></a>
<FONT color="green">277</FONT>            }<a name="line.277"></a>
<FONT color="green">278</FONT>    <a name="line.278"></a>
<FONT color="green">279</FONT>            // divide by k!<a name="line.279"></a>
<FONT color="green">280</FONT>            for (int i = 2; i &lt;= k; i++) {<a name="line.280"></a>
<FONT color="green">281</FONT>                logSum -= FastMath.log(i);<a name="line.281"></a>
<FONT color="green">282</FONT>            }<a name="line.282"></a>
<FONT color="green">283</FONT>    <a name="line.283"></a>
<FONT color="green">284</FONT>            return logSum;<a name="line.284"></a>
<FONT color="green">285</FONT>        }<a name="line.285"></a>
<FONT color="green">286</FONT>    <a name="line.286"></a>
<FONT color="green">287</FONT>        /**<a name="line.287"></a>
<FONT color="green">288</FONT>         * Returns n!. Shorthand for {@code n} &lt;a<a name="line.288"></a>
<FONT color="green">289</FONT>         * href="http://mathworld.wolfram.com/Factorial.html"&gt; Factorial&lt;/a&gt;, the<a name="line.289"></a>
<FONT color="green">290</FONT>         * product of the numbers {@code 1,...,n}.<a name="line.290"></a>
<FONT color="green">291</FONT>         * &lt;p&gt;<a name="line.291"></a>
<FONT color="green">292</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.292"></a>
<FONT color="green">293</FONT>         * &lt;ul&gt;<a name="line.293"></a>
<FONT color="green">294</FONT>         * &lt;li&gt; {@code n &gt;= 0} (otherwise<a name="line.294"></a>
<FONT color="green">295</FONT>         * {@code IllegalArgumentException} is thrown)&lt;/li&gt;<a name="line.295"></a>
<FONT color="green">296</FONT>         * &lt;li&gt; The result is small enough to fit into a {@code long}. The<a name="line.296"></a>
<FONT color="green">297</FONT>         * largest value of {@code n} for which {@code n!} &lt;<a name="line.297"></a>
<FONT color="green">298</FONT>         * Long.MAX_VALUE} is 20. If the computed value exceeds {@code Long.MAX_VALUE}<a name="line.298"></a>
<FONT color="green">299</FONT>         * an {@code ArithMeticException } is thrown.&lt;/li&gt;<a name="line.299"></a>
<FONT color="green">300</FONT>         * &lt;/ul&gt;<a name="line.300"></a>
<FONT color="green">301</FONT>         * &lt;/p&gt;<a name="line.301"></a>
<FONT color="green">302</FONT>         *<a name="line.302"></a>
<FONT color="green">303</FONT>         * @param n argument<a name="line.303"></a>
<FONT color="green">304</FONT>         * @return {@code n!}<a name="line.304"></a>
<FONT color="green">305</FONT>         * @throws MathArithmeticException if the result is too large to be represented<a name="line.305"></a>
<FONT color="green">306</FONT>         * by a {@code long}.<a name="line.306"></a>
<FONT color="green">307</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.307"></a>
<FONT color="green">308</FONT>         * @throws MathArithmeticException if {@code n &gt; 20}: The factorial value is too<a name="line.308"></a>
<FONT color="green">309</FONT>         * large to fit in a {@code long}.<a name="line.309"></a>
<FONT color="green">310</FONT>         */<a name="line.310"></a>
<FONT color="green">311</FONT>        public static long factorial(final int n) throws NotPositiveException, MathArithmeticException {<a name="line.311"></a>
<FONT color="green">312</FONT>            if (n &lt; 0) {<a name="line.312"></a>
<FONT color="green">313</FONT>                throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,<a name="line.313"></a>
<FONT color="green">314</FONT>                                               n);<a name="line.314"></a>
<FONT color="green">315</FONT>            }<a name="line.315"></a>
<FONT color="green">316</FONT>            if (n &gt; 20) {<a name="line.316"></a>
<FONT color="green">317</FONT>                throw new MathArithmeticException();<a name="line.317"></a>
<FONT color="green">318</FONT>            }<a name="line.318"></a>
<FONT color="green">319</FONT>            return FACTORIALS[n];<a name="line.319"></a>
<FONT color="green">320</FONT>        }<a name="line.320"></a>
<FONT color="green">321</FONT>    <a name="line.321"></a>
<FONT color="green">322</FONT>        /**<a name="line.322"></a>
<FONT color="green">323</FONT>         * Compute n!, the&lt;a href="http://mathworld.wolfram.com/Factorial.html"&gt;<a name="line.323"></a>
<FONT color="green">324</FONT>         * factorial&lt;/a&gt; of {@code n} (the product of the numbers 1 to n), as a<a name="line.324"></a>
<FONT color="green">325</FONT>         * {@code double}.<a name="line.325"></a>
<FONT color="green">326</FONT>         * The result should be small enough to fit into a {@code double}: The<a name="line.326"></a>
<FONT color="green">327</FONT>         * largest {@code n} for which {@code n! &lt; Double.MAX_VALUE} is 170.<a name="line.327"></a>
<FONT color="green">328</FONT>         * If the computed value exceeds {@code Double.MAX_VALUE},<a name="line.328"></a>
<FONT color="green">329</FONT>         * {@code Double.POSITIVE_INFINITY} is returned.<a name="line.329"></a>
<FONT color="green">330</FONT>         *<a name="line.330"></a>
<FONT color="green">331</FONT>         * @param n Argument.<a name="line.331"></a>
<FONT color="green">332</FONT>         * @return {@code n!}<a name="line.332"></a>
<FONT color="green">333</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.333"></a>
<FONT color="green">334</FONT>         */<a name="line.334"></a>
<FONT color="green">335</FONT>        public static double factorialDouble(final int n) throws NotPositiveException {<a name="line.335"></a>
<FONT color="green">336</FONT>            if (n &lt; 0) {<a name="line.336"></a>
<FONT color="green">337</FONT>                throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,<a name="line.337"></a>
<FONT color="green">338</FONT>                                               n);<a name="line.338"></a>
<FONT color="green">339</FONT>            }<a name="line.339"></a>
<FONT color="green">340</FONT>            if (n &lt; 21) {<a name="line.340"></a>
<FONT color="green">341</FONT>                return FACTORIALS[n];<a name="line.341"></a>
<FONT color="green">342</FONT>            }<a name="line.342"></a>
<FONT color="green">343</FONT>            return FastMath.floor(FastMath.exp(ArithmeticUtils.factorialLog(n)) + 0.5);<a name="line.343"></a>
<FONT color="green">344</FONT>        }<a name="line.344"></a>
<FONT color="green">345</FONT>    <a name="line.345"></a>
<FONT color="green">346</FONT>        /**<a name="line.346"></a>
<FONT color="green">347</FONT>         * Compute the natural logarithm of the factorial of {@code n}.<a name="line.347"></a>
<FONT color="green">348</FONT>         *<a name="line.348"></a>
<FONT color="green">349</FONT>         * @param n Argument.<a name="line.349"></a>
<FONT color="green">350</FONT>         * @return {@code n!}<a name="line.350"></a>
<FONT color="green">351</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.351"></a>
<FONT color="green">352</FONT>         */<a name="line.352"></a>
<FONT color="green">353</FONT>        public static double factorialLog(final int n) throws NotPositiveException {<a name="line.353"></a>
<FONT color="green">354</FONT>            if (n &lt; 0) {<a name="line.354"></a>
<FONT color="green">355</FONT>                throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,<a name="line.355"></a>
<FONT color="green">356</FONT>                                               n);<a name="line.356"></a>
<FONT color="green">357</FONT>            }<a name="line.357"></a>
<FONT color="green">358</FONT>            if (n &lt; 21) {<a name="line.358"></a>
<FONT color="green">359</FONT>                return FastMath.log(FACTORIALS[n]);<a name="line.359"></a>
<FONT color="green">360</FONT>            }<a name="line.360"></a>
<FONT color="green">361</FONT>            double logSum = 0;<a name="line.361"></a>
<FONT color="green">362</FONT>            for (int i = 2; i &lt;= n; i++) {<a name="line.362"></a>
<FONT color="green">363</FONT>                logSum += FastMath.log(i);<a name="line.363"></a>
<FONT color="green">364</FONT>            }<a name="line.364"></a>
<FONT color="green">365</FONT>            return logSum;<a name="line.365"></a>
<FONT color="green">366</FONT>        }<a name="line.366"></a>
<FONT color="green">367</FONT>    <a name="line.367"></a>
<FONT color="green">368</FONT>        /**<a name="line.368"></a>
<FONT color="green">369</FONT>         * Computes the greatest common divisor of the absolute value of two<a name="line.369"></a>
<FONT color="green">370</FONT>         * numbers, using a modified version of the "binary gcd" method.<a name="line.370"></a>
<FONT color="green">371</FONT>         * See Knuth 4.5.2 algorithm B.<a name="line.371"></a>
<FONT color="green">372</FONT>         * The algorithm is due to Josef Stein (1961).<a name="line.372"></a>
<FONT color="green">373</FONT>         * &lt;br/&gt;<a name="line.373"></a>
<FONT color="green">374</FONT>         * Special cases:<a name="line.374"></a>
<FONT color="green">375</FONT>         * &lt;ul&gt;<a name="line.375"></a>
<FONT color="green">376</FONT>         *  &lt;li&gt;The invocations<a name="line.376"></a>
<FONT color="green">377</FONT>         *   {@code gcd(Integer.MIN_VALUE, Integer.MIN_VALUE)},<a name="line.377"></a>
<FONT color="green">378</FONT>         *   {@code gcd(Integer.MIN_VALUE, 0)} and<a name="line.378"></a>
<FONT color="green">379</FONT>         *   {@code gcd(0, Integer.MIN_VALUE)} throw an<a name="line.379"></a>
<FONT color="green">380</FONT>         *   {@code ArithmeticException}, because the result would be 2^31, which<a name="line.380"></a>
<FONT color="green">381</FONT>         *   is too large for an int value.&lt;/li&gt;<a name="line.381"></a>
<FONT color="green">382</FONT>         *  &lt;li&gt;The result of {@code gcd(x, x)}, {@code gcd(0, x)} and<a name="line.382"></a>
<FONT color="green">383</FONT>         *   {@code gcd(x, 0)} is the absolute value of {@code x}, except<a name="line.383"></a>
<FONT color="green">384</FONT>         *   for the special cases above.&lt;/li&gt;<a name="line.384"></a>
<FONT color="green">385</FONT>         *  &lt;li&gt;The invocation {@code gcd(0, 0)} is the only one which returns<a name="line.385"></a>
<FONT color="green">386</FONT>         *   {@code 0}.&lt;/li&gt;<a name="line.386"></a>
<FONT color="green">387</FONT>         * &lt;/ul&gt;<a name="line.387"></a>
<FONT color="green">388</FONT>         *<a name="line.388"></a>
<FONT color="green">389</FONT>         * @param p Number.<a name="line.389"></a>
<FONT color="green">390</FONT>         * @param q Number.<a name="line.390"></a>
<FONT color="green">391</FONT>         * @return the greatest common divisor (never negative).<a name="line.391"></a>
<FONT color="green">392</FONT>         * @throws MathArithmeticException if the result cannot be represented as<a name="line.392"></a>
<FONT color="green">393</FONT>         * a non-negative {@code int} value.<a name="line.393"></a>
<FONT color="green">394</FONT>         * @since 1.1<a name="line.394"></a>
<FONT color="green">395</FONT>         */<a name="line.395"></a>
<FONT color="green">396</FONT>        public static int gcd(int p,<a name="line.396"></a>
<FONT color="green">397</FONT>                              int q)<a name="line.397"></a>
<FONT color="green">398</FONT>            throws MathArithmeticException {<a name="line.398"></a>
<FONT color="green">399</FONT>            int a = p;<a name="line.399"></a>
<FONT color="green">400</FONT>            int b = q;<a name="line.400"></a>
<FONT color="green">401</FONT>            if (a == 0 ||<a name="line.401"></a>
<FONT color="green">402</FONT>                b == 0) {<a name="line.402"></a>
<FONT color="green">403</FONT>                if (a == Integer.MIN_VALUE ||<a name="line.403"></a>
<FONT color="green">404</FONT>                    b == Integer.MIN_VALUE) {<a name="line.404"></a>
<FONT color="green">405</FONT>                    throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,<a name="line.405"></a>
<FONT color="green">406</FONT>                                                      p, q);<a name="line.406"></a>
<FONT color="green">407</FONT>                }<a name="line.407"></a>
<FONT color="green">408</FONT>                return FastMath.abs(a + b);<a name="line.408"></a>
<FONT color="green">409</FONT>            }<a name="line.409"></a>
<FONT color="green">410</FONT>    <a name="line.410"></a>
<FONT color="green">411</FONT>            long al = a;<a name="line.411"></a>
<FONT color="green">412</FONT>            long bl = b;<a name="line.412"></a>
<FONT color="green">413</FONT>            boolean useLong = false;<a name="line.413"></a>
<FONT color="green">414</FONT>            if (a &lt; 0) {<a name="line.414"></a>
<FONT color="green">415</FONT>                if(Integer.MIN_VALUE == a) {<a name="line.415"></a>
<FONT color="green">416</FONT>                    useLong = true;<a name="line.416"></a>
<FONT color="green">417</FONT>                } else {<a name="line.417"></a>
<FONT color="green">418</FONT>                    a = -a;<a name="line.418"></a>
<FONT color="green">419</FONT>                }<a name="line.419"></a>
<FONT color="green">420</FONT>                al = -al;<a name="line.420"></a>
<FONT color="green">421</FONT>            }<a name="line.421"></a>
<FONT color="green">422</FONT>            if (b &lt; 0) {<a name="line.422"></a>
<FONT color="green">423</FONT>                if (Integer.MIN_VALUE == b) {<a name="line.423"></a>
<FONT color="green">424</FONT>                    useLong = true;<a name="line.424"></a>
<FONT color="green">425</FONT>                } else {<a name="line.425"></a>
<FONT color="green">426</FONT>                    b = -b;<a name="line.426"></a>
<FONT color="green">427</FONT>                }<a name="line.427"></a>
<FONT color="green">428</FONT>                bl = -bl;<a name="line.428"></a>
<FONT color="green">429</FONT>            }<a name="line.429"></a>
<FONT color="green">430</FONT>            if (useLong) {<a name="line.430"></a>
<FONT color="green">431</FONT>                if(al == bl) {<a name="line.431"></a>
<FONT color="green">432</FONT>                    throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,<a name="line.432"></a>
<FONT color="green">433</FONT>                                                      p, q);<a name="line.433"></a>
<FONT color="green">434</FONT>                }<a name="line.434"></a>
<FONT color="green">435</FONT>                long blbu = bl;<a name="line.435"></a>
<FONT color="green">436</FONT>                bl = al;<a name="line.436"></a>
<FONT color="green">437</FONT>                al = blbu % al;<a name="line.437"></a>
<FONT color="green">438</FONT>                if (al == 0) {<a name="line.438"></a>
<FONT color="green">439</FONT>                    if (bl &gt; Integer.MAX_VALUE) {<a name="line.439"></a>
<FONT color="green">440</FONT>                        throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,<a name="line.440"></a>
<FONT color="green">441</FONT>                                                          p, q);<a name="line.441"></a>
<FONT color="green">442</FONT>                    }<a name="line.442"></a>
<FONT color="green">443</FONT>                    return (int) bl;<a name="line.443"></a>
<FONT color="green">444</FONT>                }<a name="line.444"></a>
<FONT color="green">445</FONT>                blbu = bl;<a name="line.445"></a>
<FONT color="green">446</FONT>    <a name="line.446"></a>
<FONT color="green">447</FONT>                // Now "al" and "bl" fit in an "int".<a name="line.447"></a>
<FONT color="green">448</FONT>                b = (int) al;<a name="line.448"></a>
<FONT color="green">449</FONT>                a = (int) (blbu % al);<a name="line.449"></a>
<FONT color="green">450</FONT>            }<a name="line.450"></a>
<FONT color="green">451</FONT>    <a name="line.451"></a>
<FONT color="green">452</FONT>            return gcdPositive(a, b);<a name="line.452"></a>
<FONT color="green">453</FONT>        }<a name="line.453"></a>
<FONT color="green">454</FONT>    <a name="line.454"></a>
<FONT color="green">455</FONT>        /**<a name="line.455"></a>
<FONT color="green">456</FONT>         * Computes the greatest common divisor of two &lt;em&gt;positive&lt;/em&gt; numbers<a name="line.456"></a>
<FONT color="green">457</FONT>         * (this precondition is &lt;em&gt;not&lt;/em&gt; checked and the result is undefined<a name="line.457"></a>
<FONT color="green">458</FONT>         * if not fulfilled) using the "binary gcd" method which avoids division<a name="line.458"></a>
<FONT color="green">459</FONT>         * and modulo operations.<a name="line.459"></a>
<FONT color="green">460</FONT>         * See Knuth 4.5.2 algorithm B.<a name="line.460"></a>
<FONT color="green">461</FONT>         * The algorithm is due to Josef Stein (1961).<a name="line.461"></a>
<FONT color="green">462</FONT>         * &lt;br/&gt;<a name="line.462"></a>
<FONT color="green">463</FONT>         * Special cases:<a name="line.463"></a>
<FONT color="green">464</FONT>         * &lt;ul&gt;<a name="line.464"></a>
<FONT color="green">465</FONT>         *  &lt;li&gt;The result of {@code gcd(x, x)}, {@code gcd(0, x)} and<a name="line.465"></a>
<FONT color="green">466</FONT>         *   {@code gcd(x, 0)} is the value of {@code x}.&lt;/li&gt;<a name="line.466"></a>
<FONT color="green">467</FONT>         *  &lt;li&gt;The invocation {@code gcd(0, 0)} is the only one which returns<a name="line.467"></a>
<FONT color="green">468</FONT>         *   {@code 0}.&lt;/li&gt;<a name="line.468"></a>
<FONT color="green">469</FONT>         * &lt;/ul&gt;<a name="line.469"></a>
<FONT color="green">470</FONT>         *<a name="line.470"></a>
<FONT color="green">471</FONT>         * @param a Positive number.<a name="line.471"></a>
<FONT color="green">472</FONT>         * @param b Positive number.<a name="line.472"></a>
<FONT color="green">473</FONT>         * @return the greatest common divisor.<a name="line.473"></a>
<FONT color="green">474</FONT>         */<a name="line.474"></a>
<FONT color="green">475</FONT>        private static int gcdPositive(int a,<a name="line.475"></a>
<FONT color="green">476</FONT>                                       int b) {<a name="line.476"></a>
<FONT color="green">477</FONT>            if (a == 0) {<a name="line.477"></a>
<FONT color="green">478</FONT>                return b;<a name="line.478"></a>
<FONT color="green">479</FONT>            }<a name="line.479"></a>
<FONT color="green">480</FONT>            else if (b == 0) {<a name="line.480"></a>
<FONT color="green">481</FONT>                return a;<a name="line.481"></a>
<FONT color="green">482</FONT>            }<a name="line.482"></a>
<FONT color="green">483</FONT>    <a name="line.483"></a>
<FONT color="green">484</FONT>            // Make "a" and "b" odd, keeping track of common power of 2.<a name="line.484"></a>
<FONT color="green">485</FONT>            final int aTwos = Integer.numberOfTrailingZeros(a);<a name="line.485"></a>
<FONT color="green">486</FONT>            a &gt;&gt;= aTwos;<a name="line.486"></a>
<FONT color="green">487</FONT>            final int bTwos = Integer.numberOfTrailingZeros(b);<a name="line.487"></a>
<FONT color="green">488</FONT>            b &gt;&gt;= bTwos;<a name="line.488"></a>
<FONT color="green">489</FONT>            final int shift = Math.min(aTwos, bTwos);<a name="line.489"></a>
<FONT color="green">490</FONT>    <a name="line.490"></a>
<FONT color="green">491</FONT>            // "a" and "b" are positive.<a name="line.491"></a>
<FONT color="green">492</FONT>            // If a &gt; b then "gdc(a, b)" is equal to "gcd(a - b, b)".<a name="line.492"></a>
<FONT color="green">493</FONT>            // If a &lt; b then "gcd(a, b)" is equal to "gcd(b - a, a)".<a name="line.493"></a>
<FONT color="green">494</FONT>            // Hence, in the successive iterations:<a name="line.494"></a>
<FONT color="green">495</FONT>            //  "a" becomes the absolute difference of the current values,<a name="line.495"></a>
<FONT color="green">496</FONT>            //  "b" becomes the minimum of the current values.<a name="line.496"></a>
<FONT color="green">497</FONT>            while (a != b) {<a name="line.497"></a>
<FONT color="green">498</FONT>                final int delta = a - b;<a name="line.498"></a>
<FONT color="green">499</FONT>                b = Math.min(a, b);<a name="line.499"></a>
<FONT color="green">500</FONT>                a = Math.abs(delta);<a name="line.500"></a>
<FONT color="green">501</FONT>    <a name="line.501"></a>
<FONT color="green">502</FONT>                // Remove any power of 2 in "a" ("b" is guaranteed to be odd).<a name="line.502"></a>
<FONT color="green">503</FONT>                a &gt;&gt;= Integer.numberOfTrailingZeros(a);<a name="line.503"></a>
<FONT color="green">504</FONT>            }<a name="line.504"></a>
<FONT color="green">505</FONT>    <a name="line.505"></a>
<FONT color="green">506</FONT>            // Recover the common power of 2.<a name="line.506"></a>
<FONT color="green">507</FONT>            return a &lt;&lt; shift;<a name="line.507"></a>
<FONT color="green">508</FONT>        }<a name="line.508"></a>
<FONT color="green">509</FONT>    <a name="line.509"></a>
<FONT color="green">510</FONT>        /**<a name="line.510"></a>
<FONT color="green">511</FONT>         * &lt;p&gt;<a name="line.511"></a>
<FONT color="green">512</FONT>         * Gets the greatest common divisor of the absolute value of two numbers,<a name="line.512"></a>
<FONT color="green">513</FONT>         * using the "binary gcd" method which avoids division and modulo<a name="line.513"></a>
<FONT color="green">514</FONT>         * operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef<a name="line.514"></a>
<FONT color="green">515</FONT>         * Stein (1961).<a name="line.515"></a>
<FONT color="green">516</FONT>         * &lt;/p&gt;<a name="line.516"></a>
<FONT color="green">517</FONT>         * Special cases:<a name="line.517"></a>
<FONT color="green">518</FONT>         * &lt;ul&gt;<a name="line.518"></a>
<FONT color="green">519</FONT>         * &lt;li&gt;The invocations<a name="line.519"></a>
<FONT color="green">520</FONT>         * {@code gcd(Long.MIN_VALUE, Long.MIN_VALUE)},<a name="line.520"></a>
<FONT color="green">521</FONT>         * {@code gcd(Long.MIN_VALUE, 0L)} and<a name="line.521"></a>
<FONT color="green">522</FONT>         * {@code gcd(0L, Long.MIN_VALUE)} throw an<a name="line.522"></a>
<FONT color="green">523</FONT>         * {@code ArithmeticException}, because the result would be 2^63, which<a name="line.523"></a>
<FONT color="green">524</FONT>         * is too large for a long value.&lt;/li&gt;<a name="line.524"></a>
<FONT color="green">525</FONT>         * &lt;li&gt;The result of {@code gcd(x, x)}, {@code gcd(0L, x)} and<a name="line.525"></a>
<FONT color="green">526</FONT>         * {@code gcd(x, 0L)} is the absolute value of {@code x}, except<a name="line.526"></a>
<FONT color="green">527</FONT>         * for the special cases above.<a name="line.527"></a>
<FONT color="green">528</FONT>         * &lt;li&gt;The invocation {@code gcd(0L, 0L)} is the only one which returns<a name="line.528"></a>
<FONT color="green">529</FONT>         * {@code 0L}.&lt;/li&gt;<a name="line.529"></a>
<FONT color="green">530</FONT>         * &lt;/ul&gt;<a name="line.530"></a>
<FONT color="green">531</FONT>         *<a name="line.531"></a>
<FONT color="green">532</FONT>         * @param p Number.<a name="line.532"></a>
<FONT color="green">533</FONT>         * @param q Number.<a name="line.533"></a>
<FONT color="green">534</FONT>         * @return the greatest common divisor, never negative.<a name="line.534"></a>
<FONT color="green">535</FONT>         * @throws MathArithmeticException if the result cannot be represented as<a name="line.535"></a>
<FONT color="green">536</FONT>         * a non-negative {@code long} value.<a name="line.536"></a>
<FONT color="green">537</FONT>         * @since 2.1<a name="line.537"></a>
<FONT color="green">538</FONT>         */<a name="line.538"></a>
<FONT color="green">539</FONT>        public static long gcd(final long p, final long q) throws MathArithmeticException {<a name="line.539"></a>
<FONT color="green">540</FONT>            long u = p;<a name="line.540"></a>
<FONT color="green">541</FONT>            long v = q;<a name="line.541"></a>
<FONT color="green">542</FONT>            if ((u == 0) || (v == 0)) {<a name="line.542"></a>
<FONT color="green">543</FONT>                if ((u == Long.MIN_VALUE) || (v == Long.MIN_VALUE)){<a name="line.543"></a>
<FONT color="green">544</FONT>                    throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS,<a name="line.544"></a>
<FONT color="green">545</FONT>                                                      p, q);<a name="line.545"></a>
<FONT color="green">546</FONT>                }<a name="line.546"></a>
<FONT color="green">547</FONT>                return FastMath.abs(u) + FastMath.abs(v);<a name="line.547"></a>
<FONT color="green">548</FONT>            }<a name="line.548"></a>
<FONT color="green">549</FONT>            // keep u and v negative, as negative integers range down to<a name="line.549"></a>
<FONT color="green">550</FONT>            // -2^63, while positive numbers can only be as large as 2^63-1<a name="line.550"></a>
<FONT color="green">551</FONT>            // (i.e. we can't necessarily negate a negative number without<a name="line.551"></a>
<FONT color="green">552</FONT>            // overflow)<a name="line.552"></a>
<FONT color="green">553</FONT>            /* assert u!=0 &amp;&amp; v!=0; */<a name="line.553"></a>
<FONT color="green">554</FONT>            if (u &gt; 0) {<a name="line.554"></a>
<FONT color="green">555</FONT>                u = -u;<a name="line.555"></a>
<FONT color="green">556</FONT>            } // make u negative<a name="line.556"></a>
<FONT color="green">557</FONT>            if (v &gt; 0) {<a name="line.557"></a>
<FONT color="green">558</FONT>                v = -v;<a name="line.558"></a>
<FONT color="green">559</FONT>            } // make v negative<a name="line.559"></a>
<FONT color="green">560</FONT>            // B1. [Find power of 2]<a name="line.560"></a>
<FONT color="green">561</FONT>            int k = 0;<a name="line.561"></a>
<FONT color="green">562</FONT>            while ((u &amp; 1) == 0 &amp;&amp; (v &amp; 1) == 0 &amp;&amp; k &lt; 63) { // while u and v are<a name="line.562"></a>
<FONT color="green">563</FONT>                                                                // both even...<a name="line.563"></a>
<FONT color="green">564</FONT>                u /= 2;<a name="line.564"></a>
<FONT color="green">565</FONT>                v /= 2;<a name="line.565"></a>
<FONT color="green">566</FONT>                k++; // cast out twos.<a name="line.566"></a>
<FONT color="green">567</FONT>            }<a name="line.567"></a>
<FONT color="green">568</FONT>            if (k == 63) {<a name="line.568"></a>
<FONT color="green">569</FONT>                throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS,<a name="line.569"></a>
<FONT color="green">570</FONT>                                                  p, q);<a name="line.570"></a>
<FONT color="green">571</FONT>            }<a name="line.571"></a>
<FONT color="green">572</FONT>            // B2. Initialize: u and v have been divided by 2^k and at least<a name="line.572"></a>
<FONT color="green">573</FONT>            // one is odd.<a name="line.573"></a>
<FONT color="green">574</FONT>            long t = ((u &amp; 1) == 1) ? v : -(u / 2)/* B3 */;<a name="line.574"></a>
<FONT color="green">575</FONT>            // t negative: u was odd, v may be even (t replaces v)<a name="line.575"></a>
<FONT color="green">576</FONT>            // t positive: u was even, v is odd (t replaces u)<a name="line.576"></a>
<FONT color="green">577</FONT>            do {<a name="line.577"></a>
<FONT color="green">578</FONT>                /* assert u&lt;0 &amp;&amp; v&lt;0; */<a name="line.578"></a>
<FONT color="green">579</FONT>                // B4/B3: cast out twos from t.<a name="line.579"></a>
<FONT color="green">580</FONT>                while ((t &amp; 1) == 0) { // while t is even..<a name="line.580"></a>
<FONT color="green">581</FONT>                    t /= 2; // cast out twos<a name="line.581"></a>
<FONT color="green">582</FONT>                }<a name="line.582"></a>
<FONT color="green">583</FONT>                // B5 [reset max(u,v)]<a name="line.583"></a>
<FONT color="green">584</FONT>                if (t &gt; 0) {<a name="line.584"></a>
<FONT color="green">585</FONT>                    u = -t;<a name="line.585"></a>
<FONT color="green">586</FONT>                } else {<a name="line.586"></a>
<FONT color="green">587</FONT>                    v = t;<a name="line.587"></a>
<FONT color="green">588</FONT>                }<a name="line.588"></a>
<FONT color="green">589</FONT>                // B6/B3. at this point both u and v should be odd.<a name="line.589"></a>
<FONT color="green">590</FONT>                t = (v - u) / 2;<a name="line.590"></a>
<FONT color="green">591</FONT>                // |u| larger: t positive (replace u)<a name="line.591"></a>
<FONT color="green">592</FONT>                // |v| larger: t negative (replace v)<a name="line.592"></a>
<FONT color="green">593</FONT>            } while (t != 0);<a name="line.593"></a>
<FONT color="green">594</FONT>            return -u * (1L &lt;&lt; k); // gcd is u*2^k<a name="line.594"></a>
<FONT color="green">595</FONT>        }<a name="line.595"></a>
<FONT color="green">596</FONT>    <a name="line.596"></a>
<FONT color="green">597</FONT>        /**<a name="line.597"></a>
<FONT color="green">598</FONT>         * &lt;p&gt;<a name="line.598"></a>
<FONT color="green">599</FONT>         * Returns the least common multiple of the absolute value of two numbers,<a name="line.599"></a>
<FONT color="green">600</FONT>         * using the formula {@code lcm(a,b) = (a / gcd(a,b)) * b}.<a name="line.600"></a>
<FONT color="green">601</FONT>         * &lt;/p&gt;<a name="line.601"></a>
<FONT color="green">602</FONT>         * Special cases:<a name="line.602"></a>
<FONT color="green">603</FONT>         * &lt;ul&gt;<a name="line.603"></a>
<FONT color="green">604</FONT>         * &lt;li&gt;The invocations {@code lcm(Integer.MIN_VALUE, n)} and<a name="line.604"></a>
<FONT color="green">605</FONT>         * {@code lcm(n, Integer.MIN_VALUE)}, where {@code abs(n)} is a<a name="line.605"></a>
<FONT color="green">606</FONT>         * power of 2, throw an {@code ArithmeticException}, because the result<a name="line.606"></a>
<FONT color="green">607</FONT>         * would be 2^31, which is too large for an int value.&lt;/li&gt;<a name="line.607"></a>
<FONT color="green">608</FONT>         * &lt;li&gt;The result of {@code lcm(0, x)} and {@code lcm(x, 0)} is<a name="line.608"></a>
<FONT color="green">609</FONT>         * {@code 0} for any {@code x}.<a name="line.609"></a>
<FONT color="green">610</FONT>         * &lt;/ul&gt;<a name="line.610"></a>
<FONT color="green">611</FONT>         *<a name="line.611"></a>
<FONT color="green">612</FONT>         * @param a Number.<a name="line.612"></a>
<FONT color="green">613</FONT>         * @param b Number.<a name="line.613"></a>
<FONT color="green">614</FONT>         * @return the least common multiple, never negative.<a name="line.614"></a>
<FONT color="green">615</FONT>         * @throws MathArithmeticException if the result cannot be represented as<a name="line.615"></a>
<FONT color="green">616</FONT>         * a non-negative {@code int} value.<a name="line.616"></a>
<FONT color="green">617</FONT>         * @since 1.1<a name="line.617"></a>
<FONT color="green">618</FONT>         */<a name="line.618"></a>
<FONT color="green">619</FONT>        public static int lcm(int a, int b) throws MathArithmeticException {<a name="line.619"></a>
<FONT color="green">620</FONT>            if (a == 0 || b == 0){<a name="line.620"></a>
<FONT color="green">621</FONT>                return 0;<a name="line.621"></a>
<FONT color="green">622</FONT>            }<a name="line.622"></a>
<FONT color="green">623</FONT>            int lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));<a name="line.623"></a>
<FONT color="green">624</FONT>            if (lcm == Integer.MIN_VALUE) {<a name="line.624"></a>
<FONT color="green">625</FONT>                throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_32_BITS,<a name="line.625"></a>
<FONT color="green">626</FONT>                                                  a, b);<a name="line.626"></a>
<FONT color="green">627</FONT>            }<a name="line.627"></a>
<FONT color="green">628</FONT>            return lcm;<a name="line.628"></a>
<FONT color="green">629</FONT>        }<a name="line.629"></a>
<FONT color="green">630</FONT>    <a name="line.630"></a>
<FONT color="green">631</FONT>        /**<a name="line.631"></a>
<FONT color="green">632</FONT>         * &lt;p&gt;<a name="line.632"></a>
<FONT color="green">633</FONT>         * Returns the least common multiple of the absolute value of two numbers,<a name="line.633"></a>
<FONT color="green">634</FONT>         * using the formula {@code lcm(a,b) = (a / gcd(a,b)) * b}.<a name="line.634"></a>
<FONT color="green">635</FONT>         * &lt;/p&gt;<a name="line.635"></a>
<FONT color="green">636</FONT>         * Special cases:<a name="line.636"></a>
<FONT color="green">637</FONT>         * &lt;ul&gt;<a name="line.637"></a>
<FONT color="green">638</FONT>         * &lt;li&gt;The invocations {@code lcm(Long.MIN_VALUE, n)} and<a name="line.638"></a>
<FONT color="green">639</FONT>         * {@code lcm(n, Long.MIN_VALUE)}, where {@code abs(n)} is a<a name="line.639"></a>
<FONT color="green">640</FONT>         * power of 2, throw an {@code ArithmeticException}, because the result<a name="line.640"></a>
<FONT color="green">641</FONT>         * would be 2^63, which is too large for an int value.&lt;/li&gt;<a name="line.641"></a>
<FONT color="green">642</FONT>         * &lt;li&gt;The result of {@code lcm(0L, x)} and {@code lcm(x, 0L)} is<a name="line.642"></a>
<FONT color="green">643</FONT>         * {@code 0L} for any {@code x}.<a name="line.643"></a>
<FONT color="green">644</FONT>         * &lt;/ul&gt;<a name="line.644"></a>
<FONT color="green">645</FONT>         *<a name="line.645"></a>
<FONT color="green">646</FONT>         * @param a Number.<a name="line.646"></a>
<FONT color="green">647</FONT>         * @param b Number.<a name="line.647"></a>
<FONT color="green">648</FONT>         * @return the least common multiple, never negative.<a name="line.648"></a>
<FONT color="green">649</FONT>         * @throws MathArithmeticException if the result cannot be represented<a name="line.649"></a>
<FONT color="green">650</FONT>         * as a non-negative {@code long} value.<a name="line.650"></a>
<FONT color="green">651</FONT>         * @since 2.1<a name="line.651"></a>
<FONT color="green">652</FONT>         */<a name="line.652"></a>
<FONT color="green">653</FONT>        public static long lcm(long a, long b) throws MathArithmeticException {<a name="line.653"></a>
<FONT color="green">654</FONT>            if (a == 0 || b == 0){<a name="line.654"></a>
<FONT color="green">655</FONT>                return 0;<a name="line.655"></a>
<FONT color="green">656</FONT>            }<a name="line.656"></a>
<FONT color="green">657</FONT>            long lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));<a name="line.657"></a>
<FONT color="green">658</FONT>            if (lcm == Long.MIN_VALUE){<a name="line.658"></a>
<FONT color="green">659</FONT>                throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_64_BITS,<a name="line.659"></a>
<FONT color="green">660</FONT>                                                  a, b);<a name="line.660"></a>
<FONT color="green">661</FONT>            }<a name="line.661"></a>
<FONT color="green">662</FONT>            return lcm;<a name="line.662"></a>
<FONT color="green">663</FONT>        }<a name="line.663"></a>
<FONT color="green">664</FONT>    <a name="line.664"></a>
<FONT color="green">665</FONT>        /**<a name="line.665"></a>
<FONT color="green">666</FONT>         * Multiply two integers, checking for overflow.<a name="line.666"></a>
<FONT color="green">667</FONT>         *<a name="line.667"></a>
<FONT color="green">668</FONT>         * @param x Factor.<a name="line.668"></a>
<FONT color="green">669</FONT>         * @param y Factor.<a name="line.669"></a>
<FONT color="green">670</FONT>         * @return the product {@code x * y}.<a name="line.670"></a>
<FONT color="green">671</FONT>         * @throws MathArithmeticException if the result can not be<a name="line.671"></a>
<FONT color="green">672</FONT>         * represented as an {@code int}.<a name="line.672"></a>
<FONT color="green">673</FONT>         * @since 1.1<a name="line.673"></a>
<FONT color="green">674</FONT>         */<a name="line.674"></a>
<FONT color="green">675</FONT>        public static int mulAndCheck(int x, int y) throws MathArithmeticException {<a name="line.675"></a>
<FONT color="green">676</FONT>            long m = ((long)x) * ((long)y);<a name="line.676"></a>
<FONT color="green">677</FONT>            if (m &lt; Integer.MIN_VALUE || m &gt; Integer.MAX_VALUE) {<a name="line.677"></a>
<FONT color="green">678</FONT>                throw new MathArithmeticException();<a name="line.678"></a>
<FONT color="green">679</FONT>            }<a name="line.679"></a>
<FONT color="green">680</FONT>            return (int)m;<a name="line.680"></a>
<FONT color="green">681</FONT>        }<a name="line.681"></a>
<FONT color="green">682</FONT>    <a name="line.682"></a>
<FONT color="green">683</FONT>        /**<a name="line.683"></a>
<FONT color="green">684</FONT>         * Multiply two long integers, checking for overflow.<a name="line.684"></a>
<FONT color="green">685</FONT>         *<a name="line.685"></a>
<FONT color="green">686</FONT>         * @param a Factor.<a name="line.686"></a>
<FONT color="green">687</FONT>         * @param b Factor.<a name="line.687"></a>
<FONT color="green">688</FONT>         * @return the product {@code a * b}.<a name="line.688"></a>
<FONT color="green">689</FONT>         * @throws MathArithmeticException if the result can not be represented<a name="line.689"></a>
<FONT color="green">690</FONT>         * as a {@code long}.<a name="line.690"></a>
<FONT color="green">691</FONT>         * @since 1.2<a name="line.691"></a>
<FONT color="green">692</FONT>         */<a name="line.692"></a>
<FONT color="green">693</FONT>        public static long mulAndCheck(long a, long b) throws MathArithmeticException {<a name="line.693"></a>
<FONT color="green">694</FONT>            long ret;<a name="line.694"></a>
<FONT color="green">695</FONT>            if (a &gt; b) {<a name="line.695"></a>
<FONT color="green">696</FONT>                // use symmetry to reduce boundary cases<a name="line.696"></a>
<FONT color="green">697</FONT>                ret = mulAndCheck(b, a);<a name="line.697"></a>
<FONT color="green">698</FONT>            } else {<a name="line.698"></a>
<FONT color="green">699</FONT>                if (a &lt; 0) {<a name="line.699"></a>
<FONT color="green">700</FONT>                    if (b &lt; 0) {<a name="line.700"></a>
<FONT color="green">701</FONT>                        // check for positive overflow with negative a, negative b<a name="line.701"></a>
<FONT color="green">702</FONT>                        if (a &gt;= Long.MAX_VALUE / b) {<a name="line.702"></a>
<FONT color="green">703</FONT>                            ret = a * b;<a name="line.703"></a>
<FONT color="green">704</FONT>                        } else {<a name="line.704"></a>
<FONT color="green">705</FONT>                            throw new MathArithmeticException();<a name="line.705"></a>
<FONT color="green">706</FONT>                        }<a name="line.706"></a>
<FONT color="green">707</FONT>                    } else if (b &gt; 0) {<a name="line.707"></a>
<FONT color="green">708</FONT>                        // check for negative overflow with negative a, positive b<a name="line.708"></a>
<FONT color="green">709</FONT>                        if (Long.MIN_VALUE / b &lt;= a) {<a name="line.709"></a>
<FONT color="green">710</FONT>                            ret = a * b;<a name="line.710"></a>
<FONT color="green">711</FONT>                        } else {<a name="line.711"></a>
<FONT color="green">712</FONT>                            throw new MathArithmeticException();<a name="line.712"></a>
<FONT color="green">713</FONT>    <a name="line.713"></a>
<FONT color="green">714</FONT>                        }<a name="line.714"></a>
<FONT color="green">715</FONT>                    } else {<a name="line.715"></a>
<FONT color="green">716</FONT>                        // assert b == 0<a name="line.716"></a>
<FONT color="green">717</FONT>                        ret = 0;<a name="line.717"></a>
<FONT color="green">718</FONT>                    }<a name="line.718"></a>
<FONT color="green">719</FONT>                } else if (a &gt; 0) {<a name="line.719"></a>
<FONT color="green">720</FONT>                    // assert a &gt; 0<a name="line.720"></a>
<FONT color="green">721</FONT>                    // assert b &gt; 0<a name="line.721"></a>
<FONT color="green">722</FONT>    <a name="line.722"></a>
<FONT color="green">723</FONT>                    // check for positive overflow with positive a, positive b<a name="line.723"></a>
<FONT color="green">724</FONT>                    if (a &lt;= Long.MAX_VALUE / b) {<a name="line.724"></a>
<FONT color="green">725</FONT>                        ret = a * b;<a name="line.725"></a>
<FONT color="green">726</FONT>                    } else {<a name="line.726"></a>
<FONT color="green">727</FONT>                        throw new MathArithmeticException();<a name="line.727"></a>
<FONT color="green">728</FONT>                    }<a name="line.728"></a>
<FONT color="green">729</FONT>                } else {<a name="line.729"></a>
<FONT color="green">730</FONT>                    // assert a == 0<a name="line.730"></a>
<FONT color="green">731</FONT>                    ret = 0;<a name="line.731"></a>
<FONT color="green">732</FONT>                }<a name="line.732"></a>
<FONT color="green">733</FONT>            }<a name="line.733"></a>
<FONT color="green">734</FONT>            return ret;<a name="line.734"></a>
<FONT color="green">735</FONT>        }<a name="line.735"></a>
<FONT color="green">736</FONT>    <a name="line.736"></a>
<FONT color="green">737</FONT>        /**<a name="line.737"></a>
<FONT color="green">738</FONT>         * Subtract two integers, checking for overflow.<a name="line.738"></a>
<FONT color="green">739</FONT>         *<a name="line.739"></a>
<FONT color="green">740</FONT>         * @param x Minuend.<a name="line.740"></a>
<FONT color="green">741</FONT>         * @param y Subtrahend.<a name="line.741"></a>
<FONT color="green">742</FONT>         * @return the difference {@code x - y}.<a name="line.742"></a>
<FONT color="green">743</FONT>         * @throws MathArithmeticException if the result can not be represented<a name="line.743"></a>
<FONT color="green">744</FONT>         * as an {@code int}.<a name="line.744"></a>
<FONT color="green">745</FONT>         * @since 1.1<a name="line.745"></a>
<FONT color="green">746</FONT>         */<a name="line.746"></a>
<FONT color="green">747</FONT>        public static int subAndCheck(int x, int y) throws MathArithmeticException {<a name="line.747"></a>
<FONT color="green">748</FONT>            long s = (long)x - (long)y;<a name="line.748"></a>
<FONT color="green">749</FONT>            if (s &lt; Integer.MIN_VALUE || s &gt; Integer.MAX_VALUE) {<a name="line.749"></a>
<FONT color="green">750</FONT>                throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_SUBTRACTION, x, y);<a name="line.750"></a>
<FONT color="green">751</FONT>            }<a name="line.751"></a>
<FONT color="green">752</FONT>            return (int)s;<a name="line.752"></a>
<FONT color="green">753</FONT>        }<a name="line.753"></a>
<FONT color="green">754</FONT>    <a name="line.754"></a>
<FONT color="green">755</FONT>        /**<a name="line.755"></a>
<FONT color="green">756</FONT>         * Subtract two long integers, checking for overflow.<a name="line.756"></a>
<FONT color="green">757</FONT>         *<a name="line.757"></a>
<FONT color="green">758</FONT>         * @param a Value.<a name="line.758"></a>
<FONT color="green">759</FONT>         * @param b Value.<a name="line.759"></a>
<FONT color="green">760</FONT>         * @return the difference {@code a - b}.<a name="line.760"></a>
<FONT color="green">761</FONT>         * @throws MathArithmeticException if the result can not be represented as a<a name="line.761"></a>
<FONT color="green">762</FONT>         * {@code long}.<a name="line.762"></a>
<FONT color="green">763</FONT>         * @since 1.2<a name="line.763"></a>
<FONT color="green">764</FONT>         */<a name="line.764"></a>
<FONT color="green">765</FONT>        public static long subAndCheck(long a, long b) throws MathArithmeticException {<a name="line.765"></a>
<FONT color="green">766</FONT>            long ret;<a name="line.766"></a>
<FONT color="green">767</FONT>            if (b == Long.MIN_VALUE) {<a name="line.767"></a>
<FONT color="green">768</FONT>                if (a &lt; 0) {<a name="line.768"></a>
<FONT color="green">769</FONT>                    ret = a - b;<a name="line.769"></a>
<FONT color="green">770</FONT>                } else {<a name="line.770"></a>
<FONT color="green">771</FONT>                    throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, a, -b);<a name="line.771"></a>
<FONT color="green">772</FONT>                }<a name="line.772"></a>
<FONT color="green">773</FONT>            } else {<a name="line.773"></a>
<FONT color="green">774</FONT>                // use additive inverse<a name="line.774"></a>
<FONT color="green">775</FONT>                ret = addAndCheck(a, -b, LocalizedFormats.OVERFLOW_IN_ADDITION);<a name="line.775"></a>
<FONT color="green">776</FONT>            }<a name="line.776"></a>
<FONT color="green">777</FONT>            return ret;<a name="line.777"></a>
<FONT color="green">778</FONT>        }<a name="line.778"></a>
<FONT color="green">779</FONT>    <a name="line.779"></a>
<FONT color="green">780</FONT>        /**<a name="line.780"></a>
<FONT color="green">781</FONT>         * Raise an int to an int power.<a name="line.781"></a>
<FONT color="green">782</FONT>         *<a name="line.782"></a>
<FONT color="green">783</FONT>         * @param k Number to raise.<a name="line.783"></a>
<FONT color="green">784</FONT>         * @param e Exponent (must be positive or zero).<a name="line.784"></a>
<FONT color="green">785</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.785"></a>
<FONT color="green">786</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.786"></a>
<FONT color="green">787</FONT>         */<a name="line.787"></a>
<FONT color="green">788</FONT>        public static int pow(final int k, int e) throws NotPositiveException {<a name="line.788"></a>
<FONT color="green">789</FONT>            if (e &lt; 0) {<a name="line.789"></a>
<FONT color="green">790</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.790"></a>
<FONT color="green">791</FONT>            }<a name="line.791"></a>
<FONT color="green">792</FONT>    <a name="line.792"></a>
<FONT color="green">793</FONT>            int result = 1;<a name="line.793"></a>
<FONT color="green">794</FONT>            int k2p    = k;<a name="line.794"></a>
<FONT color="green">795</FONT>            while (e != 0) {<a name="line.795"></a>
<FONT color="green">796</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.796"></a>
<FONT color="green">797</FONT>                    result *= k2p;<a name="line.797"></a>
<FONT color="green">798</FONT>                }<a name="line.798"></a>
<FONT color="green">799</FONT>                k2p *= k2p;<a name="line.799"></a>
<FONT color="green">800</FONT>                e = e &gt;&gt; 1;<a name="line.800"></a>
<FONT color="green">801</FONT>            }<a name="line.801"></a>
<FONT color="green">802</FONT>    <a name="line.802"></a>
<FONT color="green">803</FONT>            return result;<a name="line.803"></a>
<FONT color="green">804</FONT>        }<a name="line.804"></a>
<FONT color="green">805</FONT>    <a name="line.805"></a>
<FONT color="green">806</FONT>        /**<a name="line.806"></a>
<FONT color="green">807</FONT>         * Raise an int to a long power.<a name="line.807"></a>
<FONT color="green">808</FONT>         *<a name="line.808"></a>
<FONT color="green">809</FONT>         * @param k Number to raise.<a name="line.809"></a>
<FONT color="green">810</FONT>         * @param e Exponent (must be positive or zero).<a name="line.810"></a>
<FONT color="green">811</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.811"></a>
<FONT color="green">812</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.812"></a>
<FONT color="green">813</FONT>         */<a name="line.813"></a>
<FONT color="green">814</FONT>        public static int pow(final int k, long e) throws NotPositiveException {<a name="line.814"></a>
<FONT color="green">815</FONT>            if (e &lt; 0) {<a name="line.815"></a>
<FONT color="green">816</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.816"></a>
<FONT color="green">817</FONT>            }<a name="line.817"></a>
<FONT color="green">818</FONT>    <a name="line.818"></a>
<FONT color="green">819</FONT>            int result = 1;<a name="line.819"></a>
<FONT color="green">820</FONT>            int k2p    = k;<a name="line.820"></a>
<FONT color="green">821</FONT>            while (e != 0) {<a name="line.821"></a>
<FONT color="green">822</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.822"></a>
<FONT color="green">823</FONT>                    result *= k2p;<a name="line.823"></a>
<FONT color="green">824</FONT>                }<a name="line.824"></a>
<FONT color="green">825</FONT>                k2p *= k2p;<a name="line.825"></a>
<FONT color="green">826</FONT>                e = e &gt;&gt; 1;<a name="line.826"></a>
<FONT color="green">827</FONT>            }<a name="line.827"></a>
<FONT color="green">828</FONT>    <a name="line.828"></a>
<FONT color="green">829</FONT>            return result;<a name="line.829"></a>
<FONT color="green">830</FONT>        }<a name="line.830"></a>
<FONT color="green">831</FONT>    <a name="line.831"></a>
<FONT color="green">832</FONT>        /**<a name="line.832"></a>
<FONT color="green">833</FONT>         * Raise a long to an int power.<a name="line.833"></a>
<FONT color="green">834</FONT>         *<a name="line.834"></a>
<FONT color="green">835</FONT>         * @param k Number to raise.<a name="line.835"></a>
<FONT color="green">836</FONT>         * @param e Exponent (must be positive or zero).<a name="line.836"></a>
<FONT color="green">837</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.837"></a>
<FONT color="green">838</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.838"></a>
<FONT color="green">839</FONT>         */<a name="line.839"></a>
<FONT color="green">840</FONT>        public static long pow(final long k, int e) throws NotPositiveException {<a name="line.840"></a>
<FONT color="green">841</FONT>            if (e &lt; 0) {<a name="line.841"></a>
<FONT color="green">842</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.842"></a>
<FONT color="green">843</FONT>            }<a name="line.843"></a>
<FONT color="green">844</FONT>    <a name="line.844"></a>
<FONT color="green">845</FONT>            long result = 1l;<a name="line.845"></a>
<FONT color="green">846</FONT>            long k2p    = k;<a name="line.846"></a>
<FONT color="green">847</FONT>            while (e != 0) {<a name="line.847"></a>
<FONT color="green">848</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.848"></a>
<FONT color="green">849</FONT>                    result *= k2p;<a name="line.849"></a>
<FONT color="green">850</FONT>                }<a name="line.850"></a>
<FONT color="green">851</FONT>                k2p *= k2p;<a name="line.851"></a>
<FONT color="green">852</FONT>                e = e &gt;&gt; 1;<a name="line.852"></a>
<FONT color="green">853</FONT>            }<a name="line.853"></a>
<FONT color="green">854</FONT>    <a name="line.854"></a>
<FONT color="green">855</FONT>            return result;<a name="line.855"></a>
<FONT color="green">856</FONT>        }<a name="line.856"></a>
<FONT color="green">857</FONT>    <a name="line.857"></a>
<FONT color="green">858</FONT>        /**<a name="line.858"></a>
<FONT color="green">859</FONT>         * Raise a long to a long power.<a name="line.859"></a>
<FONT color="green">860</FONT>         *<a name="line.860"></a>
<FONT color="green">861</FONT>         * @param k Number to raise.<a name="line.861"></a>
<FONT color="green">862</FONT>         * @param e Exponent (must be positive or zero).<a name="line.862"></a>
<FONT color="green">863</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.863"></a>
<FONT color="green">864</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.864"></a>
<FONT color="green">865</FONT>         */<a name="line.865"></a>
<FONT color="green">866</FONT>        public static long pow(final long k, long e) throws NotPositiveException {<a name="line.866"></a>
<FONT color="green">867</FONT>            if (e &lt; 0) {<a name="line.867"></a>
<FONT color="green">868</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.868"></a>
<FONT color="green">869</FONT>            }<a name="line.869"></a>
<FONT color="green">870</FONT>    <a name="line.870"></a>
<FONT color="green">871</FONT>            long result = 1l;<a name="line.871"></a>
<FONT color="green">872</FONT>            long k2p    = k;<a name="line.872"></a>
<FONT color="green">873</FONT>            while (e != 0) {<a name="line.873"></a>
<FONT color="green">874</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.874"></a>
<FONT color="green">875</FONT>                    result *= k2p;<a name="line.875"></a>
<FONT color="green">876</FONT>                }<a name="line.876"></a>
<FONT color="green">877</FONT>                k2p *= k2p;<a name="line.877"></a>
<FONT color="green">878</FONT>                e = e &gt;&gt; 1;<a name="line.878"></a>
<FONT color="green">879</FONT>            }<a name="line.879"></a>
<FONT color="green">880</FONT>    <a name="line.880"></a>
<FONT color="green">881</FONT>            return result;<a name="line.881"></a>
<FONT color="green">882</FONT>        }<a name="line.882"></a>
<FONT color="green">883</FONT>    <a name="line.883"></a>
<FONT color="green">884</FONT>        /**<a name="line.884"></a>
<FONT color="green">885</FONT>         * Raise a BigInteger to an int power.<a name="line.885"></a>
<FONT color="green">886</FONT>         *<a name="line.886"></a>
<FONT color="green">887</FONT>         * @param k Number to raise.<a name="line.887"></a>
<FONT color="green">888</FONT>         * @param e Exponent (must be positive or zero).<a name="line.888"></a>
<FONT color="green">889</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.889"></a>
<FONT color="green">890</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.890"></a>
<FONT color="green">891</FONT>         */<a name="line.891"></a>
<FONT color="green">892</FONT>        public static BigInteger pow(final BigInteger k, int e) throws NotPositiveException {<a name="line.892"></a>
<FONT color="green">893</FONT>            if (e &lt; 0) {<a name="line.893"></a>
<FONT color="green">894</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.894"></a>
<FONT color="green">895</FONT>            }<a name="line.895"></a>
<FONT color="green">896</FONT>    <a name="line.896"></a>
<FONT color="green">897</FONT>            return k.pow(e);<a name="line.897"></a>
<FONT color="green">898</FONT>        }<a name="line.898"></a>
<FONT color="green">899</FONT>    <a name="line.899"></a>
<FONT color="green">900</FONT>        /**<a name="line.900"></a>
<FONT color="green">901</FONT>         * Raise a BigInteger to a long power.<a name="line.901"></a>
<FONT color="green">902</FONT>         *<a name="line.902"></a>
<FONT color="green">903</FONT>         * @param k Number to raise.<a name="line.903"></a>
<FONT color="green">904</FONT>         * @param e Exponent (must be positive or zero).<a name="line.904"></a>
<FONT color="green">905</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.905"></a>
<FONT color="green">906</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.906"></a>
<FONT color="green">907</FONT>         */<a name="line.907"></a>
<FONT color="green">908</FONT>        public static BigInteger pow(final BigInteger k, long e) throws NotPositiveException {<a name="line.908"></a>
<FONT color="green">909</FONT>            if (e &lt; 0) {<a name="line.909"></a>
<FONT color="green">910</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.910"></a>
<FONT color="green">911</FONT>            }<a name="line.911"></a>
<FONT color="green">912</FONT>    <a name="line.912"></a>
<FONT color="green">913</FONT>            BigInteger result = BigInteger.ONE;<a name="line.913"></a>
<FONT color="green">914</FONT>            BigInteger k2p    = k;<a name="line.914"></a>
<FONT color="green">915</FONT>            while (e != 0) {<a name="line.915"></a>
<FONT color="green">916</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.916"></a>
<FONT color="green">917</FONT>                    result = result.multiply(k2p);<a name="line.917"></a>
<FONT color="green">918</FONT>                }<a name="line.918"></a>
<FONT color="green">919</FONT>                k2p = k2p.multiply(k2p);<a name="line.919"></a>
<FONT color="green">920</FONT>                e = e &gt;&gt; 1;<a name="line.920"></a>
<FONT color="green">921</FONT>            }<a name="line.921"></a>
<FONT color="green">922</FONT>    <a name="line.922"></a>
<FONT color="green">923</FONT>            return result;<a name="line.923"></a>
<FONT color="green">924</FONT>    <a name="line.924"></a>
<FONT color="green">925</FONT>        }<a name="line.925"></a>
<FONT color="green">926</FONT>    <a name="line.926"></a>
<FONT color="green">927</FONT>        /**<a name="line.927"></a>
<FONT color="green">928</FONT>         * Raise a BigInteger to a BigInteger power.<a name="line.928"></a>
<FONT color="green">929</FONT>         *<a name="line.929"></a>
<FONT color="green">930</FONT>         * @param k Number to raise.<a name="line.930"></a>
<FONT color="green">931</FONT>         * @param e Exponent (must be positive or zero).<a name="line.931"></a>
<FONT color="green">932</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.932"></a>
<FONT color="green">933</FONT>         * @throws NotPositiveException if {@code e &lt; 0}.<a name="line.933"></a>
<FONT color="green">934</FONT>         */<a name="line.934"></a>
<FONT color="green">935</FONT>        public static BigInteger pow(final BigInteger k, BigInteger e) throws NotPositiveException {<a name="line.935"></a>
<FONT color="green">936</FONT>            if (e.compareTo(BigInteger.ZERO) &lt; 0) {<a name="line.936"></a>
<FONT color="green">937</FONT>                throw new NotPositiveException(LocalizedFormats.EXPONENT, e);<a name="line.937"></a>
<FONT color="green">938</FONT>            }<a name="line.938"></a>
<FONT color="green">939</FONT>    <a name="line.939"></a>
<FONT color="green">940</FONT>            BigInteger result = BigInteger.ONE;<a name="line.940"></a>
<FONT color="green">941</FONT>            BigInteger k2p    = k;<a name="line.941"></a>
<FONT color="green">942</FONT>            while (!BigInteger.ZERO.equals(e)) {<a name="line.942"></a>
<FONT color="green">943</FONT>                if (e.testBit(0)) {<a name="line.943"></a>
<FONT color="green">944</FONT>                    result = result.multiply(k2p);<a name="line.944"></a>
<FONT color="green">945</FONT>                }<a name="line.945"></a>
<FONT color="green">946</FONT>                k2p = k2p.multiply(k2p);<a name="line.946"></a>
<FONT color="green">947</FONT>                e = e.shiftRight(1);<a name="line.947"></a>
<FONT color="green">948</FONT>            }<a name="line.948"></a>
<FONT color="green">949</FONT>    <a name="line.949"></a>
<FONT color="green">950</FONT>            return result;<a name="line.950"></a>
<FONT color="green">951</FONT>        }<a name="line.951"></a>
<FONT color="green">952</FONT>    <a name="line.952"></a>
<FONT color="green">953</FONT>        /**<a name="line.953"></a>
<FONT color="green">954</FONT>         * Returns the &lt;a<a name="line.954"></a>
<FONT color="green">955</FONT>         * href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html"&gt;<a name="line.955"></a>
<FONT color="green">956</FONT>         * Stirling number of the second kind&lt;/a&gt;, "{@code S(n,k)}", the number of<a name="line.956"></a>
<FONT color="green">957</FONT>         * ways of partitioning an {@code n}-element set into {@code k} non-empty<a name="line.957"></a>
<FONT color="green">958</FONT>         * subsets.<a name="line.958"></a>
<FONT color="green">959</FONT>         * &lt;p&gt;<a name="line.959"></a>
<FONT color="green">960</FONT>         * The preconditions are {@code 0 &lt;= k &lt;= n } (otherwise<a name="line.960"></a>
<FONT color="green">961</FONT>         * {@code NotPositiveException} is thrown)<a name="line.961"></a>
<FONT color="green">962</FONT>         * &lt;/p&gt;<a name="line.962"></a>
<FONT color="green">963</FONT>         * @param n the size of the set<a name="line.963"></a>
<FONT color="green">964</FONT>         * @param k the number of non-empty subsets<a name="line.964"></a>
<FONT color="green">965</FONT>         * @return {@code S(n,k)}<a name="line.965"></a>
<FONT color="green">966</FONT>         * @throws NotPositiveException if {@code k &lt; 0}.<a name="line.966"></a>
<FONT color="green">967</FONT>         * @throws NumberIsTooLargeException if {@code k &gt; n}.<a name="line.967"></a>
<FONT color="green">968</FONT>         * @throws MathArithmeticException if some overflow happens, typically for n exceeding 25 and<a name="line.968"></a>
<FONT color="green">969</FONT>         * k between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)<a name="line.969"></a>
<FONT color="green">970</FONT>         * @since 3.1<a name="line.970"></a>
<FONT color="green">971</FONT>         */<a name="line.971"></a>
<FONT color="green">972</FONT>        public static long stirlingS2(final int n, final int k)<a name="line.972"></a>
<FONT color="green">973</FONT>            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {<a name="line.973"></a>
<FONT color="green">974</FONT>            if (k &lt; 0) {<a name="line.974"></a>
<FONT color="green">975</FONT>                throw new NotPositiveException(k);<a name="line.975"></a>
<FONT color="green">976</FONT>            }<a name="line.976"></a>
<FONT color="green">977</FONT>            if (k &gt; n) {<a name="line.977"></a>
<FONT color="green">978</FONT>                throw new NumberIsTooLargeException(k, n, true);<a name="line.978"></a>
<FONT color="green">979</FONT>            }<a name="line.979"></a>
<FONT color="green">980</FONT>    <a name="line.980"></a>
<FONT color="green">981</FONT>            long[][] stirlingS2 = STIRLING_S2.get();<a name="line.981"></a>
<FONT color="green">982</FONT>    <a name="line.982"></a>
<FONT color="green">983</FONT>            if (stirlingS2 == null) {<a name="line.983"></a>
<FONT color="green">984</FONT>                // the cache has never been initialized, compute the first numbers<a name="line.984"></a>
<FONT color="green">985</FONT>                // by direct recurrence relation<a name="line.985"></a>
<FONT color="green">986</FONT>    <a name="line.986"></a>
<FONT color="green">987</FONT>                // as S(26,9) = 11201516780955125625 is larger than Long.MAX_VALUE<a name="line.987"></a>
<FONT color="green">988</FONT>                // we must stop computation at row 26<a name="line.988"></a>
<FONT color="green">989</FONT>                final int maxIndex = 26;<a name="line.989"></a>
<FONT color="green">990</FONT>                stirlingS2 = new long[maxIndex][];<a name="line.990"></a>
<FONT color="green">991</FONT>                stirlingS2[0] = new long[] { 1l };<a name="line.991"></a>
<FONT color="green">992</FONT>                for (int i = 1; i &lt; stirlingS2.length; ++i) {<a name="line.992"></a>
<FONT color="green">993</FONT>                    stirlingS2[i] = new long[i + 1];<a name="line.993"></a>
<FONT color="green">994</FONT>                    stirlingS2[i][0] = 0;<a name="line.994"></a>
<FONT color="green">995</FONT>                    stirlingS2[i][1] = 1;<a name="line.995"></a>
<FONT color="green">996</FONT>                    stirlingS2[i][i] = 1;<a name="line.996"></a>
<FONT color="green">997</FONT>                    for (int j = 2; j &lt; i; ++j) {<a name="line.997"></a>
<FONT color="green">998</FONT>                        stirlingS2[i][j] = j * stirlingS2[i - 1][j] + stirlingS2[i - 1][j - 1];<a name="line.998"></a>
<FONT color="green">999</FONT>                    }<a name="line.999"></a>
<FONT color="green">1000</FONT>                }<a name="line.1000"></a>
<FONT color="green">1001</FONT>    <a name="line.1001"></a>
<FONT color="green">1002</FONT>                // atomically save the cache<a name="line.1002"></a>
<FONT color="green">1003</FONT>                STIRLING_S2.compareAndSet(null, stirlingS2);<a name="line.1003"></a>
<FONT color="green">1004</FONT>    <a name="line.1004"></a>
<FONT color="green">1005</FONT>            }<a name="line.1005"></a>
<FONT color="green">1006</FONT>    <a name="line.1006"></a>
<FONT color="green">1007</FONT>            if (n &lt; stirlingS2.length) {<a name="line.1007"></a>
<FONT color="green">1008</FONT>                // the number is in the small cache<a name="line.1008"></a>
<FONT color="green">1009</FONT>                return stirlingS2[n][k];<a name="line.1009"></a>
<FONT color="green">1010</FONT>            } else {<a name="line.1010"></a>
<FONT color="green">1011</FONT>                // use explicit formula to compute the number without caching it<a name="line.1011"></a>
<FONT color="green">1012</FONT>                if (k == 0) {<a name="line.1012"></a>
<FONT color="green">1013</FONT>                    return 0;<a name="line.1013"></a>
<FONT color="green">1014</FONT>                } else if (k == 1 || k == n) {<a name="line.1014"></a>
<FONT color="green">1015</FONT>                    return 1;<a name="line.1015"></a>
<FONT color="green">1016</FONT>                } else if (k == 2) {<a name="line.1016"></a>
<FONT color="green">1017</FONT>                    return (1l &lt;&lt; (n - 1)) - 1l;<a name="line.1017"></a>
<FONT color="green">1018</FONT>                } else if (k == n - 1) {<a name="line.1018"></a>
<FONT color="green">1019</FONT>                    return binomialCoefficient(n, 2);<a name="line.1019"></a>
<FONT color="green">1020</FONT>                } else {<a name="line.1020"></a>
<FONT color="green">1021</FONT>                    // definition formula: note that this may trigger some overflow<a name="line.1021"></a>
<FONT color="green">1022</FONT>                    long sum = 0;<a name="line.1022"></a>
<FONT color="green">1023</FONT>                    long sign = ((k &amp; 0x1) == 0) ? 1 : -1;<a name="line.1023"></a>
<FONT color="green">1024</FONT>                    for (int j = 1; j &lt;= k; ++j) {<a name="line.1024"></a>
<FONT color="green">1025</FONT>                        sign = -sign;<a name="line.1025"></a>
<FONT color="green">1026</FONT>                        sum += sign * binomialCoefficient(k, j) * pow(j, n);<a name="line.1026"></a>
<FONT color="green">1027</FONT>                        if (sum &lt; 0) {<a name="line.1027"></a>
<FONT color="green">1028</FONT>                            // there was an overflow somewhere<a name="line.1028"></a>
<FONT color="green">1029</FONT>                            throw new MathArithmeticException(LocalizedFormats.ARGUMENT_OUTSIDE_DOMAIN,<a name="line.1029"></a>
<FONT color="green">1030</FONT>                                                              n, 0, stirlingS2.length - 1);<a name="line.1030"></a>
<FONT color="green">1031</FONT>                        }<a name="line.1031"></a>
<FONT color="green">1032</FONT>                    }<a name="line.1032"></a>
<FONT color="green">1033</FONT>                    return sum / factorial(k);<a name="line.1033"></a>
<FONT color="green">1034</FONT>                }<a name="line.1034"></a>
<FONT color="green">1035</FONT>            }<a name="line.1035"></a>
<FONT color="green">1036</FONT>    <a name="line.1036"></a>
<FONT color="green">1037</FONT>        }<a name="line.1037"></a>
<FONT color="green">1038</FONT>    <a name="line.1038"></a>
<FONT color="green">1039</FONT>        /**<a name="line.1039"></a>
<FONT color="green">1040</FONT>         * Add two long integers, checking for overflow.<a name="line.1040"></a>
<FONT color="green">1041</FONT>         *<a name="line.1041"></a>
<FONT color="green">1042</FONT>         * @param a Addend.<a name="line.1042"></a>
<FONT color="green">1043</FONT>         * @param b Addend.<a name="line.1043"></a>
<FONT color="green">1044</FONT>         * @param pattern Pattern to use for any thrown exception.<a name="line.1044"></a>
<FONT color="green">1045</FONT>         * @return the sum {@code a + b}.<a name="line.1045"></a>
<FONT color="green">1046</FONT>         * @throws MathArithmeticException if the result cannot be represented<a name="line.1046"></a>
<FONT color="green">1047</FONT>         * as a {@code long}.<a name="line.1047"></a>
<FONT color="green">1048</FONT>         * @since 1.2<a name="line.1048"></a>
<FONT color="green">1049</FONT>         */<a name="line.1049"></a>
<FONT color="green">1050</FONT>         private static long addAndCheck(long a, long b, Localizable pattern) throws MathArithmeticException {<a name="line.1050"></a>
<FONT color="green">1051</FONT>            long ret;<a name="line.1051"></a>
<FONT color="green">1052</FONT>            if (a &gt; b) {<a name="line.1052"></a>
<FONT color="green">1053</FONT>                // use symmetry to reduce boundary cases<a name="line.1053"></a>
<FONT color="green">1054</FONT>                ret = addAndCheck(b, a, pattern);<a name="line.1054"></a>
<FONT color="green">1055</FONT>            } else {<a name="line.1055"></a>
<FONT color="green">1056</FONT>                // assert a &lt;= b<a name="line.1056"></a>
<FONT color="green">1057</FONT>    <a name="line.1057"></a>
<FONT color="green">1058</FONT>                if (a &lt; 0) {<a name="line.1058"></a>
<FONT color="green">1059</FONT>                    if (b &lt; 0) {<a name="line.1059"></a>
<FONT color="green">1060</FONT>                        // check for negative overflow<a name="line.1060"></a>
<FONT color="green">1061</FONT>                        if (Long.MIN_VALUE - b &lt;= a) {<a name="line.1061"></a>
<FONT color="green">1062</FONT>                            ret = a + b;<a name="line.1062"></a>
<FONT color="green">1063</FONT>                        } else {<a name="line.1063"></a>
<FONT color="green">1064</FONT>                            throw new MathArithmeticException(pattern, a, b);<a name="line.1064"></a>
<FONT color="green">1065</FONT>                        }<a name="line.1065"></a>
<FONT color="green">1066</FONT>                    } else {<a name="line.1066"></a>
<FONT color="green">1067</FONT>                        // opposite sign addition is always safe<a name="line.1067"></a>
<FONT color="green">1068</FONT>                        ret = a + b;<a name="line.1068"></a>
<FONT color="green">1069</FONT>                    }<a name="line.1069"></a>
<FONT color="green">1070</FONT>                } else {<a name="line.1070"></a>
<FONT color="green">1071</FONT>                    // assert a &gt;= 0<a name="line.1071"></a>
<FONT color="green">1072</FONT>                    // assert b &gt;= 0<a name="line.1072"></a>
<FONT color="green">1073</FONT>    <a name="line.1073"></a>
<FONT color="green">1074</FONT>                    // check for positive overflow<a name="line.1074"></a>
<FONT color="green">1075</FONT>                    if (a &lt;= Long.MAX_VALUE - b) {<a name="line.1075"></a>
<FONT color="green">1076</FONT>                        ret = a + b;<a name="line.1076"></a>
<FONT color="green">1077</FONT>                    } else {<a name="line.1077"></a>
<FONT color="green">1078</FONT>                        throw new MathArithmeticException(pattern, a, b);<a name="line.1078"></a>
<FONT color="green">1079</FONT>                    }<a name="line.1079"></a>
<FONT color="green">1080</FONT>                }<a name="line.1080"></a>
<FONT color="green">1081</FONT>            }<a name="line.1081"></a>
<FONT color="green">1082</FONT>            return ret;<a name="line.1082"></a>
<FONT color="green">1083</FONT>        }<a name="line.1083"></a>
<FONT color="green">1084</FONT>    <a name="line.1084"></a>
<FONT color="green">1085</FONT>        /**<a name="line.1085"></a>
<FONT color="green">1086</FONT>         * Check binomial preconditions.<a name="line.1086"></a>
<FONT color="green">1087</FONT>         *<a name="line.1087"></a>
<FONT color="green">1088</FONT>         * @param n Size of the set.<a name="line.1088"></a>
<FONT color="green">1089</FONT>         * @param k Size of the subsets to be counted.<a name="line.1089"></a>
<FONT color="green">1090</FONT>         * @throws NotPositiveException if {@code n &lt; 0}.<a name="line.1090"></a>
<FONT color="green">1091</FONT>         * @throws NumberIsTooLargeException if {@code k &gt; n}.<a name="line.1091"></a>
<FONT color="green">1092</FONT>         */<a name="line.1092"></a>
<FONT color="green">1093</FONT>        private static void checkBinomial(final int n, final int k) throws NumberIsTooLargeException, NotPositiveException {<a name="line.1093"></a>
<FONT color="green">1094</FONT>            if (n &lt; k) {<a name="line.1094"></a>
<FONT color="green">1095</FONT>                throw new NumberIsTooLargeException(LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,<a name="line.1095"></a>
<FONT color="green">1096</FONT>                                                    k, n, true);<a name="line.1096"></a>
<FONT color="green">1097</FONT>            }<a name="line.1097"></a>
<FONT color="green">1098</FONT>            if (n &lt; 0) {<a name="line.1098"></a>
<FONT color="green">1099</FONT>                throw new NotPositiveException(LocalizedFormats.BINOMIAL_NEGATIVE_PARAMETER, n);<a name="line.1099"></a>
<FONT color="green">1100</FONT>            }<a name="line.1100"></a>
<FONT color="green">1101</FONT>        }<a name="line.1101"></a>
<FONT color="green">1102</FONT>    <a name="line.1102"></a>
<FONT color="green">1103</FONT>        /**<a name="line.1103"></a>
<FONT color="green">1104</FONT>         * Returns true if the argument is a power of two.<a name="line.1104"></a>
<FONT color="green">1105</FONT>         *<a name="line.1105"></a>
<FONT color="green">1106</FONT>         * @param n the number to test<a name="line.1106"></a>
<FONT color="green">1107</FONT>         * @return true if the argument is a power of two<a name="line.1107"></a>
<FONT color="green">1108</FONT>         */<a name="line.1108"></a>
<FONT color="green">1109</FONT>        public static boolean isPowerOfTwo(long n) {<a name="line.1109"></a>
<FONT color="green">1110</FONT>            return (n &gt; 0) &amp;&amp; ((n &amp; (n - 1)) == 0);<a name="line.1110"></a>
<FONT color="green">1111</FONT>        }<a name="line.1111"></a>
<FONT color="green">1112</FONT>    }<a name="line.1112"></a>




























































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